5.76 Lecture #5 2/07/94 Page 1 of 10 pages. Lecture #5: Atoms: 1e and Alkali. centrifugal term ( +1)

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1 5.76 Lecture #5 /07/94 Page 1 of 10 pages 1e Atoms: H, H + e, L +, etc. coupled and uncoupled bass sets Lecture #5: Atoms: 1e and Alkal centrfugal term (+1) r radal Schrödnger Equaton spn-orbt l s r 3 n-scalng (also µ and Z) exact, nteger n and nteger Z nter-relatonshps notaton Self Consstent Feld to defne 1e orbtals: Alkal atoms (one e outsde closed shells) extenson of scalng sem-emprcal, non-nteger n* and Z eff IP E nl = R Z eff ( nl ) ( ) seems lke we have dfferent knds of correctons for the same n δ l thng. Quantum defect theory constant phase shfts along Rydberg seres propertes that probe nner vs. outer parts of orbtal penetratng vs. non-penetratng orbtals Qualtatve dfferences between 1e and alkal-lke electronc structures Patterns: Assgnment Predcton and Extrapolaton Informaton about complcated part of ψ from fudge factors 1-e Atoms Hydrogenc ( ) Ze Ze 1 H r, θφ,, s = H = + s 3 µ r µ c r ntegrate over θ,φ ψ(r,θ,φ) = R n (r)y n (θ,φ) knetc energy potental energy ncludes ( ) V () r r centrfugal barrer H nucleus-electron ψ reduces to unversal angular part Y m(θ,φ) and atom-specfc radal part R n (r) H spn-orbt heavly weghted at nucleus central force H

2 5.76 Lecture #5 /07/94 Page of 10 pages Z s charge on nucleus µ = m Nm e m N + m e s reduced mass m N = 1amu µ = m N = 00amu µ = ~ 1 part n 10 3 Bass sets: * uncoupled nm sm s + s * coupled nsm complete bass only f we nclude contnuum εsm ε > 0 H SO not dagonal n uncoupled bass because of 1 ( l + s + l s + ) [ ± = x ± y ] H SO s dagonal n coupled bass l s = 1 l ( + s ) A rgorously good Quantum Number s an egenvalue of an operator that commutes wth exact H. ĵ m ĵz l ˆl m l ˆl z s ŝ m s ŝ z note that [ z, s] 0, [s z, s] 0, but [ z, s] = 0 Snce nsm maxmally factorzes H nto 1 1 matrces, t s useful to examne the egenvalues.

3 5.76 Lecture #5 /07/94 Page 3 of 10 pages cm 1 unts H n sm nsm = δ δ δ δ hc ss m m 4 µ Z Z n = (, ) 4 + 1/ 4 En Zµ me 13.61eV n n n cm 1 small sotope shfts orbtal energes E 0 as n (E 0 at IP) 1 = 1 for m < 1 for m = 1 spn-orbt =+1/ above = 1/ normal multplet fne structure splttng decreases as n 3 ncreases as Z 4 doublet s+1 notaton e.g. P 3/ n = 0, 1,, 3, 4, 5, 6, 7 s p d f g h k P 3/ P 1/ Splttng I denotes 1st spectrum of H obey Z 4 scalng relatonshp very accurately ²E HI cm 1 L III 30 cm 1 Na XI 5400 cm 1 11th spectrum of Na Ths notaton dsagrees wth standard chemst s notaton. e.g. Sc II Sc + but an atomc spectroscopst expects SC II means Sc 1+

4 5.76 Lecture #5 /07/94 Page 4 of 10 pages Above equaton predcts exact degeneracy between n P 3/ and n D 3/. There s actually a small splttng LAMB SHIFT ~ cm 1 for H n =, = 1/, due to hgher order radatve correctons a new bass set that combnes atom and radaton feld. Beyond the scope of In the nsm or nm sm s bass sets, we can derve smple analytc expressons for matrx elements of many f(r) and f(p). These analytc expressons are explctly expressed n terms of the quantum numbers n,n,,, s,s,,, etc. For example, electronc transtons have relatve ntenstes P er f f r 1. operates only on spatal, not spn coordnates. s a vector wth respect to and [lke an angular momentum of unt length] 3. has odd party.

5 5.76 Lecture #5 /07/94 Page 5 of 10 pages we can mmedately deduce selecton rules = ±1 s = 0 (but 1 e can only have s = 1/) = 0, ±1 = 0 s possble because = l + s m s = 0 m = 0, ±1 n = any ( n = 0,1 strong because of best spatal overlap) example of formula [C&S handout] page 133 1s er np n 7 (n 1) n 5 (n + 1) n 5 n 3 example of usefulness of smple geometrc pctures = propensty rule

6 5.76 Lecture #5 /07/94 Page 6 of 10 pages s s hν try for drawng wth no change n length of s vector or angle between and s hν + 1 tends to be lengthened alternatvely, add hν to s = s hν both pctures have not sgnfcantly dfferent length than ntally OR = propensty rule

7 5.76 Lecture #5 /07/94 Page 7 of 10 pages Crucal ponts each electron orbtal a sngle (doublet) electronc state all propertes expressble as explct f(quantum numbers) wth explct Z,µ scalng establshes typcal magntudes for all observable propertes of any atom E n, IP, s o, hyperfne, transton moment, Stark effect Measurement of one property of a gven state dentfes whch state *** KEY IDEA *** t s and mples specfc predctable values for all other observable propertes of that state. Ths s what we would lke electronc structure to mean. The value of one thng s related (predctably) to another. What do we need to know about a 1e atom to know everythng? Z and µ How to fnd out? Rydberg Seres see a pattern that we expect to see n spectrum, but we stll need to assgn t. 1 3 convergence plot R 1/ [ ] 1/ ν c ν a convergence vs. a arbtrary nteger prncpal quantum number straght lne: slope 1/Z,y ntercept s b/z Alkal-lke atoms 1 e outsde of closed shells arbtrary nteger: consecutve numberng of membersofseres a = n + b unknown offset n = a - b H= H + e r separable > destroys all one-e angular momentum quantum numbers but preserves L = can t really treat 1/r as a S = s perturbaton because t s comparable n magntude to all other terms n H! J =

8 5.76 Lecture #5 /07/94 Page 8 of 10 pages Commutaton rules: H, L Bernath shows that S = 0 but J replace V (r)+ e H, 0 r by V SCF nl (r) Self Consstent Feld to defne 1e orbtals not 1e Schrödnger Equaton. [Orbtals depend on occupancy of all other orbtals.] (Best possble sngle product of N 1e orbtals.) e moves n feld defned by nucleus plus average charge dstrbuton produced by all other e. Ths s lke replacng Z n 1e Schrödnger Equaton by Z eff (r). Z at r = 0 Z eff (r) 1 at r = Represent sphercal, non-pont core by two modfcatons of scalng formulas. Z Z eff nl = Z n l S nl n l all other e n n eff n* = n δ l quantum defect, core penetraton qualtatve nterpretaton of δ * when δ > 0 n* < n net stablzaton relatve to hydrogenc orbtal wth n quantum numbers * when e n n orbtal penetrates nsde other orbtals, t sees larger Z eff (r) and s therefore stablzed. = 0 penetrates best has largest δ 1 less hardly at all non-penetratng orbtals have eff Z nl and δ = 0

9 5.76 Lecture #5 /07/94 Page 9 of 10 pages spn-orbt, hyperfne Some propertes are senstve to ampltude n the ntra-core part of an orb tal so we need Z eff Rydberg E n s, transton probabltes, Stark effect Other propertes are senstve to the long-range (extra-core) part of an orbtal so we need δ. Quantum Defect Theory the n orbtal relatve to that for n on a bare Z = 1 nucleus. δ π s a phase shft descrbes the phase of the outsde-the-core part of 0 r e /r Ze /r Z eff = 1 outsde Z eff = Z nsde actual V 0 (r) smple models V =0(eV) = 14.4Z r(å) Form of V (r) depends on because l determnes how much the valence e penetrates nto core. repulsve barrer to core penetraton: l (l + 1) + r term (centrfugal barrer)

10 5.76 Lecture #5 /07/94 Page 10 of 10 pages E nl RZ CORE n* r nl nl r nl = (n δ l ) (Z CORE = 1 for alkal atoms) a 0 [ ] 3n * l ( l +1) Z CORE Ths s much more than an emprcal correcton scheme. The quantum defect s n-ndependent. Typcal values for alkals 1.5 δ ns > δ np > δ nd > δ nf 0.